Hirota-Kimura Type Discretization of the Classical Nonholonomic Suslov Problem

TL;DR

This paper develops a Hirota-Kimura type discretization for the classical nonholonomic Suslov problem, demonstrating integrability, asymptotic behavior similar to steady rotations, and providing explicit solutions and higher-dimensional equations.

Contribution

It introduces a novel discretization method for the Suslov problem, proving integrability and analyzing asymptotic properties, with explicit solutions and extensions to higher dimensions.

Findings

Discrete trajectories tend to steady-state rotations.

The discretization preserves integrability.

Explicit solutions are derived.

Abstract

We constructed Hirota-Kimura type discretization of the classical nonholonomic Suslov problem of motion of rigid body fixed at a point. We found a first integral proving integrability. Also, we have shown that discrete trajectories asymptotically tend to a line of discrete analogies of so-called steady-state rotations. The last property completely corresponds to well-known property of the continuous Suslov case. The explicite formulae for solutions are given. In n-dimensional case we give discrete equations.